To prove the schwarz inequality they proved,they showed that the vector z is orthogonal to the vector v. If they are orthogonal, then why didn't they have written in this way : $\displaystyle \|u\|^{2}=|v\|^{2}+\|z\|^{2}$?

Another thing is that, how they got the below? I mean, I don't see the another $\|v\|^{2}$ in the below line. $$\left|{\frac {\langle u,v\rangle }{\langle v,v\rangle }}\right|^{2}\|v\|^{2}+\|z\|^{2}={\frac {|\langle u,v\rangle |^{2}}{\|v\|^{2}}}+\|z\|^{2}$$